When thermally cutting a workpiece, e.g., a metal workpiece, with a high-energy beam, e.g., a plasma beam or a laser beam, an incomplete cutting action may occur. The high-energy beam may no longer cut through the workpiece completely since an energy per unit length is not sufficient to melt a complete cutting gap volume. This may be due to an excessively high feed speed, an excessively low power of the high-energy beam, an excessively low oxygen gas pressure, or an excessively powerful defocusing of the high-energy radiation used. When the incomplete cutting action occurs, a cutting joint or a cutting gap is no longer defined over the entire workpiece thickness, and the high-energy beam may cut only a groove in the workpiece.